On Wednesday, May 5, 2021 at 2:20:31 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 5. Mai 2021 um 17:46:25 UTC+2:
> > On Wednesday, May 5, 2021 at 11:32:20 AM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 5. Mai 2021 um 00:00:05 UTC+2:
> > > > "undefinable real numbers" despite the name are not real numbers. They do not have the fundamental properties of real numbers.
> > > >
> > > Take the simple sequence (1/n). Considered as points on the real line they are existing, not created by quantifier magic like "for every n there is n+1". No. All these points are fixed and existing simultaneously. Between the points of (0, 1] and 0 there is nothing.
> > Corrrect. There is no closest point to 0 in (0,1]
> Incorrect.

Nope, trivially correct. Let x be an element of (0,1]. Then x is a real number and the real number x/2 exits (this is a property of the real numbers. Anything that does not have this property (e.g. "a dark real number") is not a real number), x_2 is an element of (0,1] and
is closer to 0 than x.

> (0,1] does not end with a gap.

Trivially correct. There is no end. 0 is a bound not an end. There is always a closer real number. It is possible for a set of real numbers to be "next" to 0 even though no element of the set is next to zero

--
William Hughes

William schrieb am Mittwoch, 5. Mai 2021 um 17:20:40 UTC+2:
> On Wednesday, May 5, 2021 at 8:25:29 AM UTC-3, WM wrote:
> > William schrieb am Dienstag, 4. Mai 2021 um 23:56:56 UTC+2:
> > > Each natural number is definable, Proof is by a trivial induction.
> > >
> > If this were true,
> It is true. The induction is trivial.
> Further note that the following three
> statements are equivalent and true for the set of natural numbers.
>
> The natural numbers are well ordered
> Induction works on the set of natural numbers
> Each natural number is the largest number of some FISON

Maybe. But then there is no complete set of natural numbers.
>
> It is easy to see that any of the three leads to a simple proof that the intersection of the endsegments is empty.

If induction works unlimited, then there are no endsegments.

> So either there are no sets without last element

That is the only alternative maintaining mathematics and in particular unlimited induction.

> or there is a result that you do not like.

It is not a matter of taste. Fact is that if |N is complete and all endsegments exist, then it is possible to empty |N step by step:

∀k ∈ ℕ: E(k+1) = E(k) \ {k}

This produces a bijection subtract(n) = E(n+1) that excludes that the empty set is realized without finite predecessors.

Have you ever wondered why CardLim can differ from LimCard?

LimCard can only be calculated from all definable cardinalities of the definable terms of the sequence. All definable endegments E(n) have cardinality ℵo. Therefore LimCard(E(n)) = ℵo.

The limit of the sequence (E(n)) is the empty set because every n has to go.. Therefore CardLim(E(n)) = 0.

But if all elements have gone and if only one element can go per step, then there must be finite terms. Contradiction. Therefore either actual infinity or matematics must be discarded - or there are dark numbers.

Regards, WM

William schrieb am Mittwoch, 5. Mai 2021 um 19:39:34 UTC+2:
> On Wednesday, May 5, 2021 at 2:20:31 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 5. Mai 2021 um 17:46:25 UTC+2:
> > > On Wednesday, May 5, 2021 at 11:32:20 AM UTC-3, WM wrote:
> > > > William schrieb am Mittwoch, 5. Mai 2021 um 00:00:05 UTC+2:
> > > > > "undefinable real numbers" despite the name are not real numbers. They do not have the fundamental properties of real numbers.
> > > > >
> > > > Take the simple sequence (1/n). Considered as points on the real line they are existing, not created by quantifier magic like "for every n there is n+1". No. All these points are fixed and existing simultaneously. Between the points of (0, 1] and 0 there is nothing.
> > > Corrrect. There is no closest point to 0 in (0,1]
> > Incorrect.
> Nope, trivially correct. Let x be an element of (0,1]. Then x is a real number and the real number x/2 exits (this is a property of the real numbers.. Anything that does not have this property (e.g. "a dark real number") is not a real number), x_2 is an element of (0,1] and
> is closer to 0 than x.

You find always the half distance. But if what you find has already been existing, then you can start with the smallest distance. If there is none, then we have not actual infinity.

Remember that in actual infinity the union of all intervals [1/(n+1), 1/n] is (0, 1] with no possible smaller point remaining. There you have covered all points.
>
> > (0,1] does not end with a gap.
>
> Trivially correct. There is no end. 0 is a bound not an end. There is always a closer real number.

Not if all intervals [1/(n+1), 1/n] are used

> It is possible for a set of real numbers to be "next" to 0 even though no element of the set is next to zero

How then can the union of all intervals [1/(n+1), 1/n] cover all points next to zero?

Regards, WM

On Wednesday, May 5, 2021 at 2:44:08 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 5. Mai 2021 um 17:20:40 UTC+2:
> > On Wednesday, May 5, 2021 at 8:25:29 AM UTC-3, WM wrote:
> > > William schrieb am Dienstag, 4. Mai 2021 um 23:56:56 UTC+2:
> > > > Each natural number is definable, Proof is by a trivial induction.
> > > >
> > > If this were true,
> > It is true. The induction is trivial.
> > Further note that the following three
> > statements are equivalent and true for the set of natural numbers.
> >
> > The natural numbers are well ordered
> > Induction works on the set of natural numbers
> > Each natural number is the largest number of some FISON

> Maybe. But then there is no complete set of natural numbers.

or there is a result that you do not like.

<skip>

> But if all elements have gone and if only one element can go per step, then there must be finite terms

Nope, Counterintuitive certainly, but it is not a contradiction.

--
William Hughes

On Wednesday, May 5, 2021 at 2:52:20 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 5. Mai 2021 um 19:39:34 UTC+2:
> > On Wednesday, May 5, 2021 at 2:20:31 PM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 5. Mai 2021 um 17:46:25 UTC+2:
> > > > On Wednesday, May 5, 2021 at 11:32:20 AM UTC-3, WM wrote:
> > > > > William schrieb am Mittwoch, 5. Mai 2021 um 00:00:05 UTC+2:
> > > > > > "undefinable real numbers" despite the name are not real numbers. They do not have the fundamental properties of real numbers.
> > > > > >
> > > > > Take the simple sequence (1/n). Considered as points on the real line they are existing, not created by quantifier magic like "for every n there is n+1". No. All these points are fixed and existing simultaneously. Between the points of (0, 1] and 0 there is nothing.
> > > > Corrrect. There is no closest point to 0 in (0,1]
> > > Incorrect.
> > Nope, trivially correct. Let x be an element of (0,1]. Then x is a real number and the real number x/2 exits (this is a property of the real numbers. Anything that does not have this property (e.g. "a dark real number") is not a real number), x_2 is an element of (0,1] and
> > is closer to 0 than x.
> You find always the half distance.
Indeed, so there is no element of (0,1] next to 0. (note that all the elements exist). (0.1] is next to 0 but no element of (0,1] is next to 0.

<skip>

> > It is possible for a set of real numbers to be "next" to 0 even though no element of the set is next to zero
> How then can the union of all intervals [1/(n+1), 1/n] cover all points next to zero?
There are no points "next" to zero (note the phase "even though no element of the set is next to zero")

--
William Hughes

William schrieb am Mittwoch, 5. Mai 2021 um 19:55:48 UTC+2:
> On Wednesday, May 5, 2021 at 2:44:08 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 5. Mai 2021 um 17:20:40 UTC+2:
> > > On Wednesday, May 5, 2021 at 8:25:29 AM UTC-3, WM wrote:
> > > > William schrieb am Dienstag, 4. Mai 2021 um 23:56:56 UTC+2:
> > > > > Each natural number is definable, Proof is by a trivial induction..
> > > > >
> > > > If this were true,
> > > It is true. The induction is trivial.
> > > Further note that the following three
> > > statements are equivalent and true for the set of natural numbers.
> > >
> > > The natural numbers are well ordered
> > > Induction works on the set of natural numbers
> > > Each natural number is the largest number of some FISON
>
> > Maybe. But then there is no complete set of natural numbers.
> or there is a result that you do not like.
> <skip>
> > But if all elements have gone and if only one element can go per step, then there must be finite terms
> Nope, Counterintuitive certainly, but it is not a contradiction.

The usual reproach that I have often encounterd from matheologians. Hopefully it is recognized by objective readers that this is a dishonest lie because mathematical definitions are not intuition.

Here is the mathematics: If |N is complete then all endsegments exist. Then it is possible to empty |N step by step:

∀k ∈ ℕ: E(k+1) = E(k) \ {k}

This produces a bijection subtract(n) = E(n+1) that excludes that the empty set is realized without finite predecessors.

Regards, WM

William schrieb am Mittwoch, 5. Mai 2021 um 20:06:33 UTC+2:

> > > It is possible for a set of real numbers to be "next" to 0 even though no element of the set is next to zero

No, that is impossible. The set is only there where its points are.

> > How then can the union of all intervals [1/(n+1), 1/n] cover all points next to zero?
> There are no points "next" to zero

Then consider the first points which are existing and following upon those which are not existing in your opinion. Somewhat must be existing.

Regards, WM

On Wednesday, May 5, 2021 at 3:20:21 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 5. Mai 2021 um 19:55:48 UTC+2:
> > On Wednesday, May 5, 2021 at 2:44:08 PM UTC-3, WM wrote:

> > > But if all elements have gone and if only one element can go per step, then there must be finite terms
> > Nope, Counterintuitive certainly, but it is not a contradiction.
> The usual reproach that I have often encounterd

And correctly. A contradiction is not something you do not like.

--
William Hughes

On Wednesday, May 5, 2021 at 3:20:28 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 5. Mai 2021 um 20:06:33 UTC+2:
>
> > > > It is possible for a set of real numbers to be "next" to 0 even though no element of the set is next to zero
> No, that is impossible.

Nope, it is a result you do not like. A set, S, of real numbers is "next" to zero if there is nothing between the set S and 0. Such a number would have to be closer to 0 than any element of S. If S does not have a closet element to 0 there is no such number, so S is "next" to zero. On the other hand every element of S is a real number, so no element of S is next to 0..

--
William Hughes

(I unintentionally posted this to you (WM) privately. Sorry.)

On 5/4/2021 3:12 PM, WM wrote:
> Jim Burns schrieb
> am Dienstag, 4. Mai 2021 um 18:21:00 UTC+2:

>> There, we agreed, you and I, that there can be.
>>
>> An alternative here that you overlooked is that
>> unit fractions 1/n do not run out.
>
> May be, but that implies that set theory is wrong.

Your position is that my arguments are correct, but
that the arguments of "set theory" are wrong. Yes?

Consider the possibility that, where "set theory" and I
seem to you to disagree, I am only giving the reasons
(as I see them) _behind_ "set theory" saying what it says.

( The other possibility, that I have succeeded where earlier
( intellectual heavyweights have failed, by puttering about
( on the internet, I deny. It's a nice fantasy, but no more.
( I also do not have X-ray vision, and I'm not bullet-proof.
( I hope you'll take my word for that.

"Set theory", as you use the term, doesn't always use sets.

"Set theory" makes statements using variables that refer to
_one of_ some collection, and, if _valid_ is true _no matter_
_which_ individual is referred to.

For example, for the collection { 1949, 4711 }, the statement
n*n + 1949*4711 = ( 1949 + 4711 )*n
is _valid_ in this sense. n refers to _one of_ 1949, 4711
and it is true no matter which it refers to.

You (WM) mostly accept this part of "set theory", the use of
variables.

The source of difficulty seems to be the use of variables
to refer to one of _infinitely-many_ individuals.

My own position is that nothing essential changes in the
use a variable when it refers to one of _infinitely-many_
individuals.

| n*n + 1949*4711 = ( 1949 + 4711 )*n
is true if n is of one of 1949, 4711.
Given only that statement, we don't have the information
needed to answer "Which one?"
However, we have that much information. We are able to
answer some questions about n, if not "Which one?"
We know the answer to "Is n < 10000?".
We still don't know which one, but we know that.

| A completable, linear crowd 0,...,k exists such that,
| for all adjacent i,j, j = i+1.
is true if k is of one of those countable-to.
Given only that statement, we don't have the information
needed to answer "Which one?"
However, we have that much information. We are able to
answer some questions about k, if not "Which one?"
We know the answer to "Is indcuction valid for k?"
We still don't know which one, but we know that.

| A completable, linear crowd 0,...,k exists such that,
| for all adjacent i,j, j = i+1.
is valid for the countable-to -- because that is
_what I mean_ by "countable-to". It's as valid with
respect to the countable-to as
| ABC is a triangle.
is valid with respect to right triangles --
because _that is what we mean_ by "right triangle".

( Actual set theory -- what I would call "set theory" --
( starts with the idea of a variable that represents
( _one of_ infinitely-many and, instead of 1949, 4711 or
( countable-to numbers, uses variables to describe _sets_

> *The space* between 1 and 0 runs out undoubtedly.

Someone must change what "runs out" means in order for
that to be true.

https://en.wikipedia.org/wiki/Equivocation

WM laid this down on his screen :
> William schrieb am Mittwoch, 5. Mai 2021 um 17:20:40 UTC+2:
>> On Wednesday, May 5, 2021 at 8:25:29 AM UTC-3, WM wrote:
>>> William schrieb am Dienstag, 4. Mai 2021 um 23:56:56 UTC+2:
>>>> Each natural number is definable, Proof is by a trivial induction.
>>>>
>>> If this were true,
>> It is true. The induction is trivial.
>> Further note that the following three
>> statements are equivalent and true for the set of natural numbers.
>>
>> The natural numbers are well ordered
>> Induction works on the set of natural numbers
>> Each natural number is the largest number of some FISON
>
> Maybe. But then there is no complete set of natural numbers.
>>
>> It is easy to see that any of the three leads to a simple proof that the
>> intersection of the endsegments is empty.
>
> If induction works unlimited, then there are no endsegments.
>
>> So either there are no sets without last element
>
> That is the only alternative maintaining mathematics and in particular
> unlimited induction.
>
>> or there is a result that you do not like.
>
> It is not a matter of taste. Fact is that if |N is complete

You still haven't looked that up have you.

William schrieb am Mittwoch, 5. Mai 2021 um 20:41:34 UTC+2:
> On Wednesday, May 5, 2021 at 3:20:28 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 5. Mai 2021 um 20:06:33 UTC+2:
> >
> > > > > It is possible for a set of real numbers to be "next" to 0 even though no element of the set is next to zero
> > No, that is impossible.
> Nope, it is a result you do not like.

Take (0, 1], it is the union of the intervals [1/(n+1), 1/n]. If you do not use one of the intervals, then its points are not covered. That shows that every interval is required and every point > 0 is covered by one of the intervals. No point > 0 is not covered including the next point x that can be addressed and all aleph_0 points between this x and 0. Therefore your statement above is null nd void. You should try to return to rational thought instead of panic utterings of nonsense.

> A set, S, of real numbers is "next" to zero if there is nothing between the set S and 0. Such a number would have to be closer to 0 than any element of S. If S does not have a closet element to 0 there is no such number, so S is "next" to zero. On the other hand every element of S is a real number, so no element of S is next to 0.

All points > 0 are covered by the union [1/(n+1), 1/n]. One of them must be the closest to 0 that you have addressed. Since no point is not covered, this next point, whatever it may be, is covered too. But between it and 0 there are many dark points.

Regards, WM